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Kernel: Python 3

ThinkDSP

This notebook contains solutions to exercises in Chapter 6: Discrete Cosine Transform

Copyright 2015 Allen Downey

License: Creative Commons Attribution 4.0 International

# Get thinkdsp.py

import os

if not os.path.exists('thinkdsp.py'):
    !wget https://github.com/AllenDowney/ThinkDSP/raw/master/code/thinkdsp.py

import numpy as np
import matplotlib.pyplot as plt

from thinkdsp import decorate

Exercise 1

In this chapter I claim that analyze1 takes time proportional to n3n^3 and analyze2 takes time proportional to n2n^2. To see if that's true, run them on a range of input sizes and time them. In IPython, you can use the magic command %timeit.

If you plot run time versus input size on a log-log scale, you should get a straight line with slope 3 for analyze1 and slope 2 for analyze2. You also might want to test dct_iv and scipy.fftpack.dct.

I'll start with a noise signal and an array of power-of-two sizes

from thinkdsp import UncorrelatedGaussianNoise

signal = UncorrelatedGaussianNoise()
noise = signal.make_wave(duration=1.0, framerate=16384)
noise.ys.shape
(16384,)

The following function takes an array of results from a timing experiment, plots the results, and fits a straight line.

from scipy.stats import linregress

loglog = dict(xscale='log', yscale='log')

def plot_bests(ns, bests):    
    plt.plot(ns, bests)
    decorate(**loglog)
    
    x = np.log(ns)
    y = np.log(bests)
    t = linregress(x,y)
    slope = t[0]

    return slope

PI2 = np.pi * 2

def analyze1(ys, fs, ts):
    """Analyze a mixture of cosines and return amplitudes.

    Works for the general case where M is not orthogonal.

    ys: wave array
    fs: frequencies in Hz
    ts: times where the signal was evaluated    

    returns: vector of amplitudes
    """
    args = np.outer(ts, fs)
    M = np.cos(PI2 * args)
    amps = np.linalg.solve(M, ys)
    return amps

def run_speed_test(ns, func):
    results = []
    for N in ns:
        print(N)
        ts = (0.5 + np.arange(N)) / N
        freqs = (0.5 + np.arange(N)) / 2
        ys = noise.ys[:N]
        result = %timeit -r1 -o func(ys, freqs, ts)
        results.append(result)
        
    bests = [result.best for result in results]
    return bests

Here are the results for analyze1.

ns = 2 ** np.arange(6, 13)
ns
array([ 64, 128, 256, 512, 1024, 2048, 4096])
bests = run_speed_test(ns, analyze1)
plot_bests(ns, bests)
64 270 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 1000 loops each) 128 834 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 1000 loops each) 256 3.63 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 100 loops each) 512 18 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 100 loops each) 1024 75.7 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 10 loops each) 2048 343 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each) 4096 1.98 s ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each)
2.1523010349686165
Image in a Jupyter notebook

The estimated slope is close to 2, not 3, as expected. One possibility is that the performance of np.linalg.solve is nearly quadratic in this range of array sizes.

Here are the results for analyze2:

def analyze2(ys, fs, ts):
    """Analyze a mixture of cosines and return amplitudes.

    Assumes that fs and ts are chosen so that M is orthogonal.

    ys: wave array
    fs: frequencies in Hz
    ts: times where the signal was evaluated    

    returns: vector of amplitudes
    """
    args = np.outer(ts, fs)
    M = np.cos(PI2 * args)
    amps = np.dot(M, ys) / 2
    return amps

bests2 = run_speed_test(ns, analyze2)
plot_bests(ns, bests2)
64 171 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 10000 loops each) 128 670 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 1000 loops each) 256 2.58 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 100 loops each) 512 10.3 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 100 loops each) 1024 45.3 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 10 loops each) 2048 165 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 10 loops each) 4096 625 ms ± 0 ns per loop (mean ± std. dev. of 1 run, 1 loop each)
1.9837162383836813
Image in a Jupyter notebook

The results for analyze2 fall in a straight line with the estimated slope close to 2, as expected.

Here are the results for the scipy.fftpack.dct

import scipy.fftpack

def scipy_dct(ys, freqs, ts):
    return scipy.fftpack.dct(ys, type=3)

bests3 = run_speed_test(ns, scipy_dct)
plot_bests(ns, bests3)
64 7.02 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 100000 loops each) 128 8.33 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 100000 loops each) 256 8.3 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 100000 loops each) 512 9.42 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 100000 loops each) 1024 13.3 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 100000 loops each) 2048 24.4 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 10000 loops each) 4096 41.9 µs ± 0 ns per loop (mean ± std. dev. of 1 run, 10000 loops each)
0.4112565850438262
Image in a Jupyter notebook

This implementation of dct is even faster. The line is curved, which means either we haven't seen the asymptotic behavior yet, or the asymptotic behavior is not a simple exponent of nn. In fact, as we'll see soon, the run time is proportional to nlog⁡nn \log n.

The following figure shows all three curves on the same axes.

plt.plot(ns, bests, label='analyze1')
plt.plot(ns, bests2, label='analyze2')
plt.plot(ns, bests3, label='fftpack.dct')
decorate(xlabel='Wave length (N)', ylabel='Time (s)', **loglog)
Image in a Jupyter notebook

Exercise 2

One of the major applications of the DCT is compression for both sound and images. In its simplest form, DCT-based compression works like this:

  1. Break a long signal into segments.

  2. Compute the DCT of each segment.

  3. Identify frequency components with amplitudes so low they are inaudible, and remove them. Store only the frequencies and amplitudes that remain.

  4. To play back the signal, load the frequencies and amplitudes for each segment and apply the inverse DCT.

Implement a version of this algorithm and apply it to a recording of music or speech. How many components can you eliminate before the difference is perceptible?

thinkdsp provides a class, Dct that is similar to a Spectrum, but which uses DCT instead of FFT.

As an example, I'll use a recording of a saxophone:

if not os.path.exists('100475__iluppai__saxophone-weep.wav'):
    !wget https://github.com/AllenDowney/ThinkDSP/raw/master/code/100475__iluppai__saxophone-weep.wav

from thinkdsp import read_wave

wave = read_wave('100475__iluppai__saxophone-weep.wav')
wave.make_audio()

Here's a short segment:

segment = wave.segment(start=1.2, duration=0.5)
segment.normalize()
segment.make_audio()

And here's the DCT of that segment:

seg_dct = segment.make_dct()
seg_dct.plot(high=4000)
decorate(xlabel='Frequency (Hz)', ylabel='DCT')
Image in a Jupyter notebook

There are only a few harmonics with substantial amplitude, and many entries near zero.

The following function takes a DCT and sets elements below thresh to 0.

def compress(dct, thresh=1):
    count = 0
    for i, amp in enumerate(dct.amps):
        if np.abs(amp) < thresh:
            dct.hs[i] = 0
            count += 1
            
    n = len(dct.amps)
    print(count, n, 100 * count / n, sep='\t')

If we apply it to the segment, we can eliminate more than 90% of the elements:

seg_dct = segment.make_dct()
compress(seg_dct, thresh=10)
seg_dct.plot(high=4000)
20457 22050 92.77551020408163
Image in a Jupyter notebook

And the result sounds the same (at least to me):

seg2 = seg_dct.make_wave()
seg2.make_audio()

To compress a longer segment, we can make a DCT spectrogram. The following function is similar to wave.make_spectrogram except that it uses the DCT.

from thinkdsp import Spectrogram

def make_dct_spectrogram(wave, seg_length):
    """Computes the DCT spectrogram of the wave.

    seg_length: number of samples in each segment

    returns: Spectrogram
    """
    window = np.hamming(seg_length)
    i, j = 0, seg_length
    step = seg_length // 2

    # map from time to Spectrum
    spec_map = {}

    while j < len(wave.ys):
        segment = wave.slice(i, j)
        segment.window(window)

        # the nominal time for this segment is the midpoint
        t = (segment.start + segment.end) / 2
        spec_map[t] = segment.make_dct()

        i += step
        j += step

    return Spectrogram(spec_map, seg_length)

Now we can make a DCT spectrogram and apply compress to each segment:

spectro = make_dct_spectrogram(wave, seg_length=1024)
for t, dct in sorted(spectro.spec_map.items()):
    compress(dct, thresh=0.2)
1018 1024 99.4140625 1016 1024 99.21875 1014 1024 99.0234375 1017 1024 99.31640625 1016 1024 99.21875 1017 1024 99.31640625 1016 1024 99.21875 1020 1024 99.609375 1014 1024 99.0234375 1005 1024 98.14453125 1009 1024 98.53515625 1015 1024 99.12109375 1015 1024 99.12109375 1016 1024 99.21875 1016 1024 99.21875 1015 1024 99.12109375 1017 1024 99.31640625 1020 1024 99.609375 1013 1024 98.92578125 1017 1024 99.31640625 1013 1024 98.92578125 1017 1024 99.31640625 1018 1024 99.4140625 1015 1024 99.12109375 1013 1024 98.92578125 794 1024 77.5390625 785 1024 76.66015625 955 1024 93.26171875 995 1024 97.16796875 992 1024 96.875 976 1024 95.3125 925 1024 90.33203125 802 1024 78.3203125 836 1024 81.640625 850 1024 83.0078125 882 1024 86.1328125 883 1024 86.23046875 891 1024 87.01171875 901 1024 87.98828125 902 1024 88.0859375 900 1024 87.890625 900 1024 87.890625 894 1024 87.3046875 904 1024 88.28125 901 1024 87.98828125 915 1024 89.35546875 913 1024 89.16015625 899 1024 87.79296875 905 1024 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1024 73.828125 756 1024 73.828125 755 1024 73.73046875 746 1024 72.8515625 756 1024 73.828125 738 1024 72.0703125 757 1024 73.92578125 764 1024 74.609375 765 1024 74.70703125 762 1024 74.4140625 768 1024 75.0 773 1024 75.48828125 782 1024 76.3671875 773 1024 75.48828125 766 1024 74.8046875 755 1024 73.73046875 766 1024 74.8046875 772 1024 75.390625 810 1024 79.1015625 739 1024 72.16796875 717 1024 70.01953125 722 1024 70.5078125 739 1024 72.16796875 725 1024 70.80078125 736 1024 71.875 759 1024 74.12109375 769 1024 75.09765625 749 1024 73.14453125 710 1024 69.3359375 748 1024 73.046875 720 1024 70.3125 732 1024 71.484375 721 1024 70.41015625 734 1024 71.6796875 763 1024 74.51171875 747 1024 72.94921875 754 1024 73.6328125 755 1024 73.73046875 764 1024 74.609375 801 1024 78.22265625 768 1024 75.0 780 1024 76.171875 773 1024 75.48828125 764 1024 74.609375 775 1024 75.68359375 740 1024 72.265625 794 1024 77.5390625 796 1024 77.734375 769 1024 75.09765625 751 1024 73.33984375 782 1024 76.3671875 758 1024 74.0234375 777 1024 75.87890625 794 1024 77.5390625 784 1024 76.5625 788 1024 76.953125 773 1024 75.48828125 783 1024 76.46484375 784 1024 76.5625 785 1024 76.66015625 806 1024 78.7109375 807 1024 78.80859375 797 1024 77.83203125 785 1024 76.66015625 794 1024 77.5390625 766 1024 74.8046875 790 1024 77.1484375 746 1024 72.8515625 762 1024 74.4140625 813 1024 79.39453125 801 1024 78.22265625 782 1024 76.3671875 776 1024 75.78125 755 1024 73.73046875 780 1024 76.171875 784 1024 76.5625 805 1024 78.61328125 791 1024 77.24609375 803 1024 78.41796875 799 1024 78.02734375 795 1024 77.63671875 797 1024 77.83203125 806 1024 78.7109375 781 1024 76.26953125 795 1024 77.63671875 797 1024 77.83203125 893 1024 87.20703125 775 1024 75.68359375 787 1024 76.85546875 746 1024 72.8515625 767 1024 74.90234375 749 1024 73.14453125 749 1024 73.14453125 738 1024 72.0703125 736 1024 71.875 747 1024 72.94921875 760 1024 74.21875 737 1024 71.97265625 752 1024 73.4375 756 1024 73.828125 772 1024 75.390625 740 1024 72.265625 737 1024 71.97265625 766 1024 74.8046875 791 1024 77.24609375 765 1024 74.70703125 771 1024 75.29296875 786 1024 76.7578125 770 1024 75.1953125 761 1024 74.31640625 765 1024 74.70703125 756 1024 73.828125 758 1024 74.0234375 765 1024 74.70703125 785 1024 76.66015625 769 1024 75.09765625 781 1024 76.26953125 792 1024 77.34375 798 1024 77.9296875 809 1024 79.00390625 778 1024 75.9765625 782 1024 76.3671875 776 1024 75.78125 791 1024 77.24609375 794 1024 77.5390625 783 1024 76.46484375 771 1024 75.29296875 792 1024 77.34375 785 1024 76.66015625 812 1024 79.296875 809 1024 79.00390625 799 1024 78.02734375 798 1024 77.9296875 803 1024 78.41796875 800 1024 78.125 805 1024 78.61328125 803 1024 78.41796875 799 1024 78.02734375 802 1024 78.3203125 804 1024 78.515625 809 1024 79.00390625 784 1024 76.5625 791 1024 77.24609375 814 1024 79.4921875 788 1024 76.953125 816 1024 79.6875 810 1024 79.1015625 820 1024 80.078125 823 1024 80.37109375 813 1024 79.39453125 799 1024 78.02734375 807 1024 78.80859375 799 1024 78.02734375 789 1024 77.05078125 813 1024 79.39453125 819 1024 79.98046875 809 1024 79.00390625 784 1024 76.5625 809 1024 79.00390625 810 1024 79.1015625 785 1024 76.66015625 838 1024 81.8359375 821 1024 80.17578125 822 1024 80.2734375 800 1024 78.125 815 1024 79.58984375 827 1024 80.76171875 820 1024 80.078125 792 1024 77.34375 818 1024 79.8828125 813 1024 79.39453125 824 1024 80.46875 795 1024 77.63671875 788 1024 76.953125 796 1024 77.734375 802 1024 78.3203125 800 1024 78.125 796 1024 77.734375 823 1024 80.37109375 804 1024 78.515625 811 1024 79.19921875 808 1024 78.90625 815 1024 79.58984375 812 1024 79.296875 822 1024 80.2734375 793 1024 77.44140625 803 1024 78.41796875 806 1024 78.7109375 812 1024 79.296875 796 1024 77.734375 804 1024 78.515625 807 1024 78.80859375 821 1024 80.17578125 793 1024 77.44140625 799 1024 78.02734375 810 1024 79.1015625 818 1024 79.8828125 813 1024 79.39453125 825 1024 80.56640625 804 1024 78.515625 821 1024 80.17578125 809 1024 79.00390625 828 1024 80.859375 813 1024 79.39453125 838 1024 81.8359375 836 1024 81.640625 818 1024 79.8828125 808 1024 78.90625 819 1024 79.98046875 820 1024 80.078125 814 1024 79.4921875 901 1024 87.98828125 894 1024 87.3046875 888 1024 86.71875 780 1024 76.171875 773 1024 75.48828125 750 1024 73.2421875 750 1024 73.2421875 730 1024 71.2890625 761 1024 74.31640625 775 1024 75.68359375 782 1024 76.3671875 788 1024 76.953125 748 1024 73.046875 752 1024 73.4375 771 1024 75.29296875 746 1024 72.8515625 778 1024 75.9765625 777 1024 75.87890625 760 1024 74.21875 734 1024 71.6796875 711 1024 69.43359375 754 1024 73.6328125 745 1024 72.75390625 758 1024 74.0234375 744 1024 72.65625 755 1024 73.73046875 749 1024 73.14453125 723 1024 70.60546875 784 1024 76.5625 761 1024 74.31640625 758 1024 74.0234375 709 1024 69.23828125 769 1024 75.09765625 773 1024 75.48828125 769 1024 75.09765625 756 1024 73.828125 747 1024 72.94921875 787 1024 76.85546875 770 1024 75.1953125 749 1024 73.14453125 769 1024 75.09765625 748 1024 73.046875 761 1024 74.31640625 759 1024 74.12109375 775 1024 75.68359375 756 1024 73.828125 774 1024 75.5859375 776 1024 75.78125 760 1024 74.21875 783 1024 76.46484375 744 1024 72.65625 766 1024 74.8046875 761 1024 74.31640625 788 1024 76.953125 774 1024 75.5859375 753 1024 73.53515625 754 1024 73.6328125 765 1024 74.70703125 736 1024 71.875 782 1024 76.3671875 768 1024 75.0 778 1024 75.9765625 767 1024 74.90234375 774 1024 75.5859375 772 1024 75.390625 769 1024 75.09765625 774 1024 75.5859375 776 1024 75.78125 796 1024 77.734375 762 1024 74.4140625 766 1024 74.8046875 765 1024 74.70703125 783 1024 76.46484375 770 1024 75.1953125 799 1024 78.02734375 779 1024 76.07421875 774 1024 75.5859375 791 1024 77.24609375 797 1024 77.83203125 781 1024 76.26953125 754 1024 73.6328125 790 1024 77.1484375 790 1024 77.1484375 801 1024 78.22265625 783 1024 76.46484375 787 1024 76.85546875 805 1024 78.61328125 758 1024 74.0234375 785 1024 76.66015625 788 1024 76.953125 806 1024 78.7109375 818 1024 79.8828125 776 1024 75.78125 807 1024 78.80859375 802 1024 78.3203125 782 1024 76.3671875 812 1024 79.296875 803 1024 78.41796875 803 1024 78.41796875 787 1024 76.85546875 799 1024 78.02734375 786 1024 76.7578125 813 1024 79.39453125 813 1024 79.39453125 813 1024 79.39453125 803 1024 78.41796875 815 1024 79.58984375 792 1024 77.34375 807 1024 78.80859375 829 1024 80.95703125 797 1024 77.83203125 814 1024 79.4921875 793 1024 77.44140625 802 1024 78.3203125 775 1024 75.68359375 816 1024 79.6875 804 1024 78.515625 808 1024 78.90625 809 1024 79.00390625 814 1024 79.4921875 808 1024 78.90625 823 1024 80.37109375 811 1024 79.19921875 806 1024 78.7109375 819 1024 79.98046875 805 1024 78.61328125 826 1024 80.6640625 826 1024 80.6640625 807 1024 78.80859375 818 1024 79.8828125 818 1024 79.8828125 812 1024 79.296875 816 1024 79.6875 815 1024 79.58984375 827 1024 80.76171875 830 1024 81.0546875 852 1024 83.203125 827 1024 80.76171875 834 1024 81.4453125 835 1024 81.54296875 835 1024 81.54296875 829 1024 80.95703125 822 1024 80.2734375 818 1024 79.8828125 827 1024 80.76171875 834 1024 81.4453125 829 1024 80.95703125 846 1024 82.6171875 829 1024 80.95703125 829 1024 80.95703125 833 1024 81.34765625 837 1024 81.73828125 837 1024 81.73828125 815 1024 79.58984375 834 1024 81.4453125 833 1024 81.34765625 840 1024 82.03125 855 1024 83.49609375 853 1024 83.30078125 853 1024 83.30078125 846 1024 82.6171875 852 1024 83.203125 856 1024 83.59375 859 1024 83.88671875 851 1024 83.10546875 845 1024 82.51953125 874 1024 85.3515625 861 1024 84.08203125 877 1024 85.64453125 853 1024 83.30078125 861 1024 84.08203125 859 1024 83.88671875 866 1024 84.5703125 868 1024 84.765625 870 1024 84.9609375 856 1024 83.59375 859 1024 83.88671875 864 1024 84.375 864 1024 84.375 876 1024 85.546875 872 1024 85.15625 872 1024 85.15625 863 1024 84.27734375 859 1024 83.88671875 878 1024 85.7421875 860 1024 83.984375 864 1024 84.375 875 1024 85.44921875 862 1024 84.1796875 867 1024 84.66796875 867 1024 84.66796875 864 1024 84.375 864 1024 84.375 876 1024 85.546875 875 1024 85.44921875 860 1024 83.984375 865 1024 84.47265625 881 1024 86.03515625 867 1024 84.66796875 869 1024 84.86328125 873 1024 85.25390625 869 1024 84.86328125 873 1024 85.25390625 873 1024 85.25390625 862 1024 84.1796875 865 1024 84.47265625 871 1024 85.05859375 869 1024 84.86328125 871 1024 85.05859375 866 1024 84.5703125 877 1024 85.64453125 861 1024 84.08203125 881 1024 86.03515625 882 1024 86.1328125 874 1024 85.3515625 875 1024 85.44921875 866 1024 84.5703125 870 1024 84.9609375 883 1024 86.23046875 870 1024 84.9609375 871 1024 85.05859375 877 1024 85.64453125 866 1024 84.5703125 877 1024 85.64453125 863 1024 84.27734375 873 1024 85.25390625 871 1024 85.05859375 883 1024 86.23046875 862 1024 84.1796875 853 1024 83.30078125 858 1024 83.7890625 857 1024 83.69140625 855 1024 83.49609375 847 1024 82.71484375 837 1024 81.73828125 850 1024 83.0078125 864 1024 84.375 879 1024 85.83984375 883 1024 86.23046875 871 1024 85.05859375 888 1024 86.71875 881 1024 86.03515625 830 1024 81.0546875 870 1024 84.9609375 877 1024 85.64453125 886 1024 86.5234375 863 1024 84.27734375 871 1024 85.05859375 886 1024 86.5234375 871 1024 85.05859375 896 1024 87.5 872 1024 85.15625 870 1024 84.9609375 877 1024 85.64453125 863 1024 84.27734375 886 1024 86.5234375 898 1024 87.6953125 884 1024 86.328125 908 1024 88.671875 878 1024 85.7421875 865 1024 84.47265625 864 1024 84.375 888 1024 86.71875 870 1024 84.9609375 862 1024 84.1796875 866 1024 84.5703125 889 1024 86.81640625 879 1024 85.83984375 884 1024 86.328125 880 1024 85.9375 876 1024 85.546875 864 1024 84.375 877 1024 85.64453125 858 1024 83.7890625 894 1024 87.3046875 890 1024 86.9140625 893 1024 87.20703125 891 1024 87.01171875 896 1024 87.5 892 1024 87.109375 906 1024 88.4765625 878 1024 85.7421875 893 1024 87.20703125 898 1024 87.6953125 888 1024 86.71875 903 1024 88.18359375 911 1024 88.96484375 911 1024 88.96484375 901 1024 87.98828125 909 1024 88.76953125 911 1024 88.96484375 921 1024 89.94140625 922 1024 90.0390625 916 1024 89.453125 923 1024 90.13671875 928 1024 90.625 920 1024 89.84375 922 1024 90.0390625 915 1024 89.35546875 930 1024 90.8203125 914 1024 89.2578125 917 1024 89.55078125 918 1024 89.6484375 921 1024 89.94140625 921 1024 89.94140625 937 1024 91.50390625 931 1024 90.91796875 923 1024 90.13671875 921 1024 89.94140625 934 1024 91.2109375 930 1024 90.8203125 933 1024 91.11328125 932 1024 91.015625 930 1024 90.8203125 930 1024 90.8203125 933 1024 91.11328125 933 1024 91.11328125 949 1024 92.67578125 941 1024 91.89453125 945 1024 92.28515625 936 1024 91.40625 956 1024 93.359375 948 1024 92.578125 936 1024 91.40625 941 1024 91.89453125 949 1024 92.67578125 941 1024 91.89453125 940 1024 91.796875 951 1024 92.87109375 941 1024 91.89453125 941 1024 91.89453125 930 1024 90.8203125 930 1024 90.8203125 924 1024 90.234375 919 1024 89.74609375 911 1024 88.96484375 934 1024 91.2109375 892 1024 87.109375 929 1024 90.72265625 922 1024 90.0390625 927 1024 90.52734375 917 1024 89.55078125 856 1024 83.59375 835 1024 81.54296875 852 1024 83.203125 870 1024 84.9609375 878 1024 85.7421875 872 1024 85.15625 894 1024 87.3046875 865 1024 84.47265625 889 1024 86.81640625 871 1024 85.05859375 873 1024 85.25390625 864 1024 84.375 859 1024 83.88671875 867 1024 84.66796875 833 1024 81.34765625 853 1024 83.30078125 874 1024 85.3515625 843 1024 82.32421875 848 1024 82.8125 844 1024 82.421875 817 1024 79.78515625 865 1024 84.47265625 807 1024 78.80859375 752 1024 73.4375 775 1024 75.68359375 772 1024 75.390625 778 1024 75.9765625 767 1024 74.90234375 784 1024 76.5625 800 1024 78.125 807 1024 78.80859375 826 1024 80.6640625 805 1024 78.61328125 788 1024 76.953125 820 1024 80.078125 809 1024 79.00390625 803 1024 78.41796875 799 1024 78.02734375 806 1024 78.7109375 839 1024 81.93359375 846 1024 82.6171875 914 1024 89.2578125 888 1024 86.71875 839 1024 81.93359375 836 1024 81.640625 830 1024 81.0546875 845 1024 82.51953125 828 1024 80.859375 834 1024 81.4453125 854 1024 83.3984375 847 1024 82.71484375 846 1024 82.6171875 845 1024 82.51953125 863 1024 84.27734375 867 1024 84.66796875 855 1024 83.49609375 844 1024 82.421875 864 1024 84.375 865 1024 84.47265625 860 1024 83.984375 868 1024 84.765625 871 1024 85.05859375 868 1024 84.765625 857 1024 83.69140625 885 1024 86.42578125 908 1024 88.671875 872 1024 85.15625 848 1024 82.8125 813 1024 79.39453125 763 1024 74.51171875 761 1024 74.31640625 779 1024 76.07421875 783 1024 76.46484375 776 1024 75.78125 784 1024 76.5625 811 1024 79.19921875 812 1024 79.296875 777 1024 75.87890625 794 1024 77.5390625 794 1024 77.5390625 813 1024 79.39453125 805 1024 78.61328125 828 1024 80.859375 796 1024 77.734375 806 1024 78.7109375 817 1024 79.78515625 840 1024 82.03125 786 1024 76.7578125 818 1024 79.8828125 832 1024 81.25 835 1024 81.54296875 768 1024 75.0 841 1024 82.12890625 833 1024 81.34765625 836 1024 81.640625 824 1024 80.46875 830 1024 81.0546875 837 1024 81.73828125 837 1024 81.73828125 858 1024 83.7890625 847 1024 82.71484375 870 1024 84.9609375 866 1024 84.5703125 843 1024 82.32421875 867 1024 84.66796875 849 1024 82.91015625 869 1024 84.86328125 860 1024 83.984375 862 1024 84.1796875 846 1024 82.6171875 854 1024 83.3984375 871 1024 85.05859375 861 1024 84.08203125 882 1024 86.1328125 892 1024 87.109375 877 1024 85.64453125 895 1024 87.40234375 887 1024 86.62109375 873 1024 85.25390625 894 1024 87.3046875 890 1024 86.9140625 878 1024 85.7421875 894 1024 87.3046875 879 1024 85.83984375 890 1024 86.9140625 895 1024 87.40234375 889 1024 86.81640625 896 1024 87.5 898 1024 87.6953125 901 1024 87.98828125 879 1024 85.83984375 890 1024 86.9140625 888 1024 86.71875 917 1024 89.55078125 902 1024 88.0859375 921 1024 89.94140625 915 1024 89.35546875 927 1024 90.52734375 923 1024 90.13671875 928 1024 90.625 923 1024 90.13671875 914 1024 89.2578125 918 1024 89.6484375 927 1024 90.52734375 926 1024 90.4296875 919 1024 89.74609375 916 1024 89.453125 928 1024 90.625 916 1024 89.453125 933 1024 91.11328125 925 1024 90.33203125 930 1024 90.8203125 930 1024 90.8203125 934 1024 91.2109375 933 1024 91.11328125 935 1024 91.30859375 939 1024 91.69921875 934 1024 91.2109375 938 1024 91.6015625 944 1024 92.1875 937 1024 91.50390625 937 1024 91.50390625 935 1024 91.30859375 937 1024 91.50390625 937 1024 91.50390625 954 1024 93.1640625 940 1024 91.796875 942 1024 91.9921875 955 1024 93.26171875 949 1024 92.67578125 941 1024 91.89453125 947 1024 92.48046875 940 1024 91.796875 943 1024 92.08984375 946 1024 92.3828125 962 1024 93.9453125 954 1024 93.1640625 956 1024 93.359375 957 1024 93.45703125 962 1024 93.9453125 960 1024 93.75 944 1024 92.1875 969 1024 94.62890625 969 1024 94.62890625 969 1024 94.62890625 968 1024 94.53125 970 1024 94.7265625 969 1024 94.62890625 975 1024 95.21484375 963 1024 94.04296875 965 1024 94.23828125 975 1024 95.21484375 969 1024 94.62890625 966 1024 94.3359375 969 1024 94.62890625 984 1024 96.09375 981 1024 95.80078125 977 1024 95.41015625 982 1024 95.8984375 980 1024 95.703125 980 1024 95.703125 981 1024 95.80078125 982 1024 95.8984375 981 1024 95.80078125 987 1024 96.38671875 982 1024 95.8984375 982 1024 95.8984375 977 1024 95.41015625 982 1024 95.8984375 986 1024 96.2890625 984 1024 96.09375 986 1024 96.2890625 987 1024 96.38671875 996 1024 97.265625 995 1024 97.16796875 997 1024 97.36328125 999 1024 97.55859375 1002 1024 97.8515625 999 1024 97.55859375 1004 1024 98.046875 1002 1024 97.8515625 999 1024 97.55859375 1000 1024 97.65625 1007 1024 98.33984375 1010 1024 98.6328125 1012 1024 98.828125 1004 1024 98.046875 1000 1024 97.65625 1008 1024 98.4375 1005 1024 98.14453125 1011 1024 98.73046875 1011 1024 98.73046875 1009 1024 98.53515625 1005 1024 98.14453125 1007 1024 98.33984375 1010 1024 98.6328125 1010 1024 98.6328125 1005 1024 98.14453125 1008 1024 98.4375 1012 1024 98.828125 1009 1024 98.53515625 1012 1024 98.828125 1013 1024 98.92578125 1010 1024 98.6328125 1012 1024 98.828125 1014 1024 99.0234375 1016 1024 99.21875 1010 1024 98.6328125 1014 1024 99.0234375 1015 1024 99.12109375 1012 1024 98.828125 1019 1024 99.51171875 1015 1024 99.12109375 1016 1024 99.21875 1019 1024 99.51171875 1019 1024 99.51171875 1016 1024 99.21875 1018 1024 99.4140625 1018 1024 99.4140625

In most segments, the compression is 75-85%.

To hear what it sounds like, we can convert the spectrogram back to a wave and play it.

wave2 = spectro.make_wave()
wave2.make_audio()

And here's the original again for comparison.

wave.make_audio()

As an experiment, you might try increasing thresh to see when the effect of compression becomes audible (to you).

Also, you might try compressing a signal with some noisy elements, like cymbals.